holomorphic curves and celestial mechanics

Publicaciones Matemáticas del UruguayVolumen 17, Julio 2019, Páginas 61–77S 0797-1443

HOLOMORPHIC CURVES AND CELESTIAL MECHANICS

UMBERTO L. HRYNIEWICZ

Abstract. This expository article has two purposes. The first is to explainthe connection between Poincaré’s work on Celestial Mechanics and the ArnoldConjecture on the minimal number of fixed points of Hamiltonian diffeomor-phisms. The second is to outline how holomorphic curve methods can be usedto study Hamiltonian dynamics.

1. From Celestial Mechanics to Arnold Conjecture

In order to reach the origins of the problems that motivated Floer Theory weneed to revisit the end of the XIX century. During this time Celestial Mechanicsserved as stage for Poincaré to implement some of the revolutionary ideas that laidthe foundations of the field of Dynamical Systems. Poincaré was led to make afundamental statement, known today as the Poincaré-Birkhoff Theorem [32], provedby Birkhoff [5] in 1913. It is a fixed point theorem for certain area-preservingannulus homeomorphisms, the precise statement will be recalled below. For decadesmathematicians looked for a better understanding of the theorem, for generalizationsto higher dimensions and to other phase spaces. We had to wait until the 1960’s whenideas of Arnold finally allowed for proper understanding and generalization. Themain purpose of this introduction is to provide more details about this remarkablechain of events.

Poincaré investigated a simplification of the 3-body problem: the planar circularrestricted 3-body problem (PCR3BP). Two massive particles evolve in a relativelycircular trajectory of the 2-body problem. A third particle (satellite) moves in thesame plane of the first two, under their influence according to Newton’s Law ofGravitation. However, the satellite is assumed not to disturb the movement ofthe massive particles. It turns out that in a rotating coordinate system, wherethe massive particles remain fixed, the satellite’s movement is described by anautonomous Hamiltonian; see Section 4 for precise formulas.

Low energy levels (below lowest critical level) have three connected components,they project onto three so-called Hill regions of the configuration plane: two of whichare bounded neighborhoods of the massive particles while the third is a neighborhoodof infinity. We choose one bounded Hill region. The mass ratio µ ∈ (0, 1) is definedas the ratio between the mass of the particle outside the chosen Hill region andthe total mass. The Hamiltonian turns out to have a smooth limit as µ→ 0. Thelimiting Hamiltonian models the rotating Kepler problem, which is just the Keplerproblem viewed in a rotating coordinate system.

The rotating Kepler problem is integrable, the integrals being the energy andthe angular momentum. The Hill region around the light particle degenerates asµ→ 0, while the boundaries of the other Hill regions converge to circles around the

c©2019 PMU

61

62 UMBERTO L. HRYNIEWICZ

limiting position of the heavy particle, which we call center. Using integrability it isnot difficult to find circular orbits of the rotating Kepler problem. There are threeof them: two inside the bounded Hill region, and a third one inside the unboundedHill region. One of the two circular orbits in the bounded Hill region is retrogradewhile the other is direct, according to which they rotate around the center in theopposite or in the same sense of the relative position of the massive particles.

Poincaré was able to find continuations of the circular orbits when µ > 0 is smallenough. For this he devised a method, called the continuation method, which wenow briefly describe. Choose a transverse section at a point of one of the orbits(recall that energy levels are three-dimensional). There is a well-defined local returnmap, and the orbit corresponds to a fixed point. The transverse section remains, ofcourse, transverse as we slightly perturb µ. If the derivative of the return map atthis fixed point is non-singular for µ = 0 then the fixed point is isolated, and theimplicit function theorem gives isolated fixed points of the local return map for allµ > 0 small enough. These correspond to continued periodic orbits. They retainthe geometric properties of the unperturbed orbits, namely one is retrograde andthe other is direct.

Now, using a method introduced by Levi-Civita [31] which we shall illustratein Section 4, collisions with the massive particle can be regularized to obtain anenergy level diffeomorphic to S3. The movement evolves smoothly there. In doingso, an ambiguity in the representation of states is introduced: the energy level isantipodal symmetric, each state being represented twice as a pair of antipodal points.We ignore this problem for now, keep the ambiguity and say that the regularizeddynamics is a two-to-one lift of the unregularized dynamics. The same regularizationprocess applies of course to the rotating Kepler problem. The direct and retrogradeorbits lift under this process to a pair of periodic orbits forming a Hopf link. Whenµ = 0 Poincaré uses the angular momentum to show that these two lifted orbitsbound a very special embedded annulus, a so-called global surface of section (GSS).

Definition 1.1. Let φt be a smooth flow on a smooth closed 3-manifold M . Aglobal surface of section for φt is an embedded compact surface S ↪→M such that

(i) The boundary ∂S consists of periodic orbits, and the interior S \ ∂S istransverse to φt.

(ii) For every p ∈M \ ∂S there exist t− < 0 < t+ such that φt±(p) ∈ S.

In the presence of a global surface of section the dynamical properties of the flowφt get encoded in the first return map

(1) ψ : S \ ∂S → S \ ∂S ψ(p) = φτ(p)(p)

where the first return time τ : S \ ∂S → (0,+∞) is defined as

(2) τ(p) = inf{t > 0 | φt(p) ∈ S}.

Poincaré showed that the annular global surface of section for µ = 0 could becontinued for µ > 0 small enough. In fact, the continuation method applies tocontinue its boundary, as explained above. Using a C∞-small isotopy, one continuesthe annulus “by hand”. The task is now to show that the continued annulus is againa global surface of section. By transversality, the return map is well-defined oncompact subsets of the interior, and globally encodes the dynamics on arbitrarilylarge compact subsets of the complement of the boundary orbits. The situation

HOLOMORPHIC CURVES AND CELESTIAL MECHANICS 63

near the boundary is more delicate, but it can be nicely controlled by the robustpositivity of the transverse rotation numbers of the boundary orbits.

A more detailed discussion about global surfaces of section is found in Section 3,where the connection to holomorphic curves will be made.

Why is it nice to have a global surface of section? Because it allows for two-dimensional methods to come into play and shed light onto the three-dimensionalflow. This is precisely what Poincaré did, and in doing so he was able to leave theseed for Symplectic Topology and Floer Theory. Let us explain this point, which ismain goal of this introduction.

Using the linearized dynamics of the direct and retrograde orbits, one can smoothlyextend the first return map up to boundary. Each boundary component is mappedonto itself. The symplectic nature of the flow allows us to find a smooth 2-form onthe annulus which is preserved by the extended return map. This 2-form defines anarea form in the interior, but vanishes on the boundary. It turns out that there is ahomeomorphism between Poincaré’s annulus and R/Z× [0, 1] that is smooth in theinterior and pulls the 2-form back to the standard area element. Such a conjugatinghomeomorphism is not differentiable on the boundary.

Thus, Poincaré was led to the problem of studying qualitative properties of area-and orientation-preserving homeomorphisms of the closed annulus that preserveboundary components. Consider such a homeomorphism f on R/Z× [0, 1]. It canbe lifted to a homeomorphism F of R × [0, 1]. If we denote its components byF (x, y) = (X(x, y), Y (x, y)) then

(3) f(x+ Z, y) = (X(x, y) + Z, Y (x, y)) ∀(x, y) ∈ R× [0, 1].

Of course, there are infinitely many choices of F , and they all differ by translationby an integer. It follows that

(4) X(x+ 1, y) = X(x, y) + 1 Y (x+ 1, y) = Y (x, y)

holds for every (x, y) ∈ R× [0, 1].Poincaré’s seminal contribution to Symplectic Dynamics and Topology was the

formulation of the following statement.

Theorem 1.2 (Poincaré’s last geometric theorem [32]). Suppose that

x 7→ X(x, 1)− x and x 7→ X(x, 0)− x

are non-vanishing functions with opposite signs. Then F has at least two fixed pointswhich are not integer translations of each other. In particular, they project ontodistinct fixed points of f .

This is a truly remarkable statement. Note that X(x, 1)− x and X(x, 0)− x are1-periodic functions of x in view of (4). The assumption that they have definite andopposite signs means that f moves boundary components in opposite directions, i.e.f “twists the annulus”.

Theorem 1.2 can be strengthened as follows. Let

πR : R× [0, 1]→ R (x, y) 7→ x

denote the projection onto the first component, and consider rotation numbers

(5) ρ0 = limn→+∞

πR ◦ Fn(x, 0)

nρ1 = lim

n→+∞

πR ◦ Fn(x, 1)

n


of F restricted to boundary components. The twist condition asks that

(6) ρ0 6= ρ1.

A different choice of F will change ρ0, ρ1 by addition of a common integer, hence (6)is really a condition on f . The assumptions of Theorem 1.2 force ρ0 and ρ1 to haveopposite signs, in particular (6) holds.

Theorem 1.3 (Poincaré-Birkhoff). If ρ0 6= ρ1 then for every p/q between ρ0 andρ1 there exist P, P ′ ∈ R× [0, 1] satisfying

F q(P ) = P + (p, 0) F q(P ′) = P ′ + (p, 0)

Moreover, the orbits of these points project to distinct periodic orbits of the map f .In particular, f admits infinitely many periodic points.

The above form of the theorem forces existence of infinitely many periodic orbitsin the PCR3BP when µ > 0 is small enough and the energy is low. This amountsfor one of the fist major victories of Poincaré’s ideas.

Birkhoff’s proof [5] did not shed light onto generalizations to higher dimensions orto different phase spaces. However, Poincaré was already able to prove Theorem 1.2in special cases, and his arguments did lend to such generalizations. This wasremarked by V. I. Arnold in the 1960’s; the following discussion is based on [3,appendix 9]. Arnold realized that, in the smooth case, Theorem 1.2 would beconsequence of a certain fixed point theorem for a special class of diffeomorphismsof the torus. Arnold’s statement can be generalized to all symplectic manifolds.This was, in fact, only one of many statements that came to be known as ArnoldConjectures. These conjectures led Floer [10, 11, 12] to develop what is nowadaysknown as Floer theory.

Arnold considered certain diffeomorphisms of the 2n-dimensional torus R2n/Z2n.The space R2n carries a standard symplectic form

ω0 =

n∑j=1

dxj ∧ dyj

where coordinates are denoted by (x1, . . . , xn, y1, . . . , yn). Since ω0 is invariantby translations, it descends to a symplectic form on R2n/Z2n again denoted byω0. Arnold considers diffeomorphisms Ψ : R2n/Z2n → R2n/Z2n with the followingproperties:(H1) Ψ is isotopic to the identity.(H2) Ψ is symplectic: Ψ∗ω0 = ω0.(H3) Ψ is exact: it admits a lift to R2n whose center of gravity vanishes.

Let us explain these in more detail. By (H1) the map Ψ lifts via the projectionR2n → R2n/Z2n to a symplectic diffeomorphism Ψ of (R2n, ω0) satisfying

(7) Ψ(z + k) = Ψ(z) + k ∀z ∈ R2n, ∀k ∈ Z2n.

Hence we can write Ψ(z) = z + ∆(z) for some Z2n-periodic function ∆. The centerof mass of Ψ is defined by the integral

(8)∫[0,1]2n

∆(z)

There are of course infinitely many lifts and they all differ by translation by a vectorin Z2n. In particular, the same holds for their centers of mass. Hence condition


(H3) is equivalent to saying that the center of mass of any lift of Ψ belongs to Z2n.We may simply refer to Ψ as an exact symplectic diffeomorphism of (R2n/Z2n, ω0).

Arnold Conjecture (Particular case): Exact symplectic diffeomorphisms of the stan-dard symplectic torus (R2n/Z2n, ω0) must have at least 2n+1 fixed points. Morevoer,if all fixed points are non-degenerate then there must be at least 22n fixed points.

To illustrate the ideas we prove the conjecture in case the exact symplecticdiffeomorphism is C1-close to the identity. Let Ψ be the unique lift to R2n that isC1-close to the identity. Identifying R2n = Rnx × Rny , we can write in components

Ψ(x, y) = (X(x, y), Y (x, y)).

The identity Ψ∗ω0 = ω0 is equivalent to saying that

η =

n∑j=1

(yj − Yj)dxj + (Xj − xj)dYj

is a closed 1-form. C1-closeness to the identity ensures that the map (x, y) 7→ (x, Y )is a diffeomorphism of R2n. Hence we may view η in (x, Y )-space and find areal-valued function F (x, Y ) such that dF = η. In other words

(9)X − x = D2F (x, Y )

y − Y = D1F (x, Y )

holds, and defines the map implicitly. Finally we make use of the vanishing of thecenter of mass to conclude that F is Z2n-periodic. Hence, by Morse theory, thefunction on R2n/Z2n induced by F has at least 2n+ 1 critical points. By (9) thismeans that Ψ has at least 2n + 1 fixed points. The non-degenerate case followssimilarly by the well-known estimates on the number of critical points of a Morsefunction in terms of sum of Betti numbers. The proof is complete.

The Arnold conjecture for standard symplectic tori was confirmed by a celebratedresult due to Conley and Zehnder [8]. Their arguments make use of degree theoryin infinite dimensions in order to take advantage of the fact that the classical actionfunctional is a compact perturbation of a quadratic form of infinite index andco-index.

The generalization to arbitrary symplectic manifolds requires the notion ofHamiltonian diffeomorphism. Let (W,ω) be a symplectic manifold and let {ψt}t∈[0,1]be a smooth isotopy of W starting at the identity ψ0 = id, consisting of symplecticdiffeomorphisms. Let Xt be the vector field generating ψt, i.e. d

dtψt = Xt ◦ ψt.

Differentiating we obtain

(10) 0 =d

dt(ψt)∗ω = (ψt)∗LXt

ω = (ψt)∗diXtω

from where we see that iXtω is closed, for every t. A diffeomorphism φ of W isotopic

to the identity is a Hamiltonian diffeomorphism of (W,ω) if it can be written asφ = ψ1 for some isotopy ψt as above such that iXtω is exact for every t. The set ofHamiltonian diffeomorphisms of (W,ω) will be denoted by Ham(W,ω).

Arnold Conjecture: Let (W,ω) be a closed symplectic manifold and let φ ∈ Ham(W,ω).Then

#Fix(φ) ≥ inf{#Crit(f) | f ∈ C∞(W,R)}


and, moreover, if all fixed points of φ are non-degenerate then

#Fix(φ) ≥∑i

bi(W )

where bi(W ) denotes the rank of Hi(W,Z).

The connection with the previously stated conjecture is that the set of exact sym-plectic diffeomorphisms of (R2n/Z2n, ω0) coincides precisely with Ham(R2n/Z2n, ω0).

To close the circle of ideas and conclude this introduction we need to explainwhy Arnold’s conjecture on (R2/Z2, ω0) implies Theorem 1.2 (in the smooth case).Consider an area and orientation preserving diffeomorphism f of R/Z× [0, 1] thatpreserves boundary components. We make a simplifying assumption for the sake ofexposition:

(∗) f is a rotation near the boundary.

If this assumption is dropped then the discussion below can be modified but thedetails would draw our attention to unimportant technical issues.

By the assumptions of Theorem 1.2 and by hypothesis (∗), there is a lift F (x, y) =(X,Y ) of f to R× [0, 1] satisfying

(11)F (x, y) = (x+ α0, y) if y ∼ 0

F (x, y) = (x+ α1, y) if y ∼ 1

where α0α1 < 0.Consider d1, d2 > 0 to be fixed a posteriori, and set d := d1 + d2. On the thicker

strip R× [0, 2 + d] consider the map Ψ defined by:

• If y ∈ [0, 1] then Ψ(x, y) = F (x, y),• If y ∈ [1, 1 + d1] then Ψ(x, y) = (x+ α1, y),• If y ∈ [1 + d1, 2 + d1] then

Ψ(x, y) = (X(x, 2 + d1 − y), 2 + d1 − Y (x, 2 + d1 − y)),

• If y ∈ [2 + d1, 2 + d] then Ψ(x, y) = (x+ α0, y).

It follows that Ψ is a diffeomorphism of R× [0, 2 + d] which agrees with the map(x, y) 7→ (x+ α0, y) near the boundary.

We extend Ψ to a map Ψ : R2 → R2 in a (2+d)-periodic fashion in the y-variable.Then the extended Ψ is an area and orientation preserving diffeomorphism of R2

that commutes with the additive action of the lattice Γ = Z× (2 + d)Z. As such, itdescends to a diffeomorphism on the torus R2/Γ.

If we write Ψ(x, y) = (x+ g(x, y), y + h(x, y)) then the center of mass of Ψ is(∫[0,1]×[0,2+d]

g(x, y),

∫[0,1]×[0,2+d]

h(x, y)

)

Now we wish to show that if d1, d2 are suitably chosen then the center of massof Ψ vanishes. To see this we first note that the second coordinate vanishes: on[0, 1]× ([1, 1 + d1] ∪ [2 + d1, 2 + d]) the function h(x, y) vanishes, and by the change


of variables formula∫[0,1]×[1+d1,2+d1]

h(x, y)

=

∫[0,1]×[1+d1,2+d1]

2 + d1 − Y (x, 2 + d1 − y)− y

=

∫[0,1]×[1+d1,2+d1]

−(Y (x, 2 + d1 − y)− (2 + d1 − y))

=

∫[0,1]×[1+d1,2+d1]

−h(x, 2 + d1 − y) = −∫[0,1]2

h(x, y).

As for the first coordinate, note that∫[0,1]×[1+d1,2+d1]

g(x, y) =

∫[0,1]2

g(x, y)

from where we get∫[0,1]×[0,2+d]

g(x, y) = α1d1 + α0d2 + 2

∫[0,1]2

g(x, y)

Since α0α1 < 0 we can choose d1, d2 positive in such a way that

α1d1 + α0d2 = −2

∫[0,1]2

g(x, y)

forcing the first coordinate of the center of mass to vanish, as desired.Hence, the validity of Arnold’s conjecture on the 2-torus will force Ψ to have at

least three fixed points. By symmetry, F has at least two fixed points, as claimedby Theorem 1.2.

2. Hamiltonian Dynamics and PDEs

An important tool for understanding Hamiltonian dynamics is the underlyingvariational structure, which incarnates in many forms. This section is devoted toexplaining the origins of the PDE methods used by Floer to explore the variationalstructure and attack the Arnold Conjecture. These methods are, of course, basedon the seminal work of Gromov [19] where holomorphic curves were first introducedas a tool to study symplectic geometry.

We start with a quick revision of the basics on symplectic geometry. Phase spacesof Hamiltonian systems are symplectic manifolds. These are pairs (W,ω) consistingof a smooth manifold W and a closed 2-form ω which is non-degenerate: at everypoint, kerω is the trivial vector space. Such forms are called symplectic forms.

Symplectic manifolds are necessarily even dimensional, but in fact more is true:locally they are just copies of open subsets of the standard symplectic vector space(R2n, ω0) where

(12) ω0 =

n∑j=1

dxj ∧ dyj .

Here the coordinates on R2n are denoted by (x1, . . . , xn, y1, . . . , yn). This is thecontent of Darboux’s theorem.

Corrections to the discussion on Floer’s chain complex and a misuse of the expression “centerof mass” have been added as erratum at the end of the volume.


Hamiltonian systems on (W,ω) are determined by (possibly time dependent) func-tions Ht : W → R as follows. The symplectic gradient, also called the Hamiltonianvector field, of Ht is the non-autonomous vector field XHt defined by

ω(XHt, ·) = dHt

for each t. The function Ht is called the Hamiltonian. In symplectic coordinatesprovided by Darboux theorem, the ODE

(13) x(t) = XHt(x(t))

is nothing but Hamilton’s equations of motion. The dynamics on (W,ω) induced bythis ODE is Hamiltonian dynamics.

When the Hamiltonian H is time independent then its values are preserved bythe flow of XH since dH(XH) = ω(XH , XH) = 0. Then H can be thought of asenergy, and the system is conservative.

The non-autonomous case. Let us assume that Ht is periodic in time, say ofperiod 1. This imposes no loss of generality, as it turns out. Consider the loop spaceL of W , defined as a space of maps R/Z→W (regularity is not specified as this isan informal discussion). If ω is exact then one considers on L the action functionalfrom classical mechanics

c 7→∫R/Z

c∗λ+

∫ 1

0

Ht(c(t)) dt

where λ is a primitive of ω. Since ω might not be exact – and it will never be whenW is closed – we need to make additional assumptions.

Assume that (W,ω) is aespherical in the sense that ω and c1(TW,ω) vanish onπ2(M). Here c1(TW,ω) denotes the first Chern class of the symplectic vector bundle(TW,ω). For the reader not familiar with Chern classes, note that its vanishingon π2(M) is equivalent to the following condition: for every map f : S2 → Wthe symplectic vector bundle f∗(TW,ω) over S2 is trivial. Let L0 ⊂ L be theconnected component consisting of contractible loops. We define

AH : L0 → R c 7→∫Dv∗ω +

∫R/Z

Ht(c(t))dt

where v : D → W is a smooth map satisfying v(ei2πt) = c(t). The aesphericityassumption guarantees that AH(c) does not depend on the choice of v. The Euler-Lagrange equations are precisely Hamilton’s equations of motion. Hence, finding(contractible) periodic motions amounts to finding critical points of AH .

Deep results in Symplectic Topology have been established by exploiting thisvariational structure. Sometimes one can define so-called symplectic capacities asspecial critical values of AH . These invariants shed new light onto the theory ofHamiltonian systems. In Gromov’s seminal work [19] one finds the first symplecticcapacity, so-called Gromov width, although the appropriate formalism was introducedlater by Ekeland and Hofer who also defined many new symplectic capacities.

There are major difficulties in the analysis of AH . Critical points have infiniteindex and co-index, in any reasonable sense. Moreover, AH is unbounded from aboveand below. If one follows any kind of anti-gradient flow of AH then critical pointsand values are expected to be missed, and sublevel sets do not change topology aswe pass critical levels.


Floer’s idea is to look at the L2 anti-gradient flow equation, even knowing that itdoes not define a flow. To be more concrete, consider ω-compatible and 1-periodicalmost complex structures Jt on W . These are 1-parameter families of fields ofendomorphisms Jt of TW such that for every t there holds Jt+1 = Jt, J2

t = −I andω(·, Jt·) is a Riemannian metric. Then

〈X,Y 〉L2 =

∫R/Z

ω(c(t))(X(t), Jt(c(t))Y (t)) dt

defines an inner-product on vector fields along c (thought of as tangent vectors ofthe loop space, based at c). The L2 anti-gradient of AH at c is the vector field

−Jt(c(t))[c(t)−XHt(c(t))]

along c. If u = u(s, t) is a map R× R/Z→W , thought of as a path of loops, thenthe anti-gradient flow equation is Floer’s equation

(14) us + Jt(u)[ut −XHt(u)] = 0.

It is a non-linear elliptic equation, a compact perturbation of the Cauchy-Riemannequation associated to J . Its analytical properties suffice to define some kind ofMorse homology of AH , called Floer homology. Note that if u(s, t) solves (14) thenu(s+ s0, t) is also a soluation, for every s0, but they will be declared equivalent forthe purpose of counting solutions.

Let us pause and describe a simple instance of Floer’s construction. By theaesphericity condition every contractible 1-periodic solution x : R/Z→W of (13)has a well-defined Conley-Zehnder index CZ(x) ∈ Z. We shall not describe theConley-Zehnder index, referring to [4] for details; we shall only say that it is relatedto winding properties of the linearization of the ODE (13).

Floer assumes that contractible 1-periodic solutions of (13) are non-degeneratein the sense that 1 is not an eigenvalue of the corresponding linearized flow. Thisis like saying that the action functional is Morse. He considers the vector spaceCF∗(H) over Z/2Z freely generated by the contractible 1-periodic solutions of (13),graded by the Conley-Zehnder index. It turns out that when Jt is generic – insome precise sense that we will not explain here – the following holds: if x, y aregenerators of CF(H) satisfying CZ(x)− CZ(y) = 1 then the number of equivalenceclasses of solutions u(s, t) of Floer’s equation (14) satisfying

lims→−∞

u(s, t) = x(t) lims→+∞

u(s, t) = y(t)

is finite. Denote the number of such equivalence classes of solutions by n(x, y). Floerdefines a degree −1 differential

(15) δH,J(x) =∑

CZ(y)=CZ(x)−1

(n(x, y) mod 2) y

on CF(H). It turns out that the homology of the chain complex (CF∗(H), δH,J)is independent of the pair (H,J) and is canonically isomorphic to the singularhomology H∗(W,Z/2Z). This explains why the sum of Betti numbers is a lowerbound on the number of contractible 1-periodic solutions of (13), at least in thenon-degenerate case.

This homology theory carries a richer structure: a filtration by action values.This fact is used to define special critical values, which in turn are used to constructinvariants. Hence the variational structure of AH sheds light onto the geometry of


the phase space. But we are also very much interested in the reverse direction: theproperties of (W,ω) might force AH to have special critical points. These periodicorbits can be used to study global properties of the underlying Hamiltonian system.For the basics in Floer homology we refer to the book [4] by Audin and Damian,and the notes [33] by Salamon.

The autonomous case. This story has a beautiful analogue for autonomoussystems, which is related to contact geometry and is due to Hofer [20]. Assume H istime-independent and look at a regular level M ⊂W . There is no loss of regularityto suppose that M = H−1(0) since we may add a constant to H without alteringthe dynamics.

Assume that ω has a primitive λ. From now on we also write λ to denote the pull-back of this primitive toM by the inclusion mapM ↪→W , with no fear of ambiguity.The crucial assumption to be made is that λ defines a contact form on M , i.e. ω ispointwise non-degenerate on the associated contact structure ξ := kerλ ⊂ TM .

The action functional on loops in M is now just∫λ. Hofer considered the

so-called symplectization. It is defined as R×M equipped with the symplectic formd(eaλ), where a denotes the R-component and we see λ as an R-invariant 1-form onR×M (with respect to the action of (R,+) on the first component).

We don’t insist in parametrizing the dynamics on M by XH . Instead, weparametrize it by the unique non-vanishing multiple Xλ of XH satisfying λ(Xλ) = 1.It is called the Reeb vector field associated to λ.

In [20] Hofer considered R-invariant almost complex structures J on R ×Mthat send ∂

∂a 7→ Xλ, and have the property that ω(·, J ·) is an inner-product on ξ.As before, we see here Xλ and ξ as R-invariant objects on R ×M . Such a J iscompatible with d(eaλ) in the sense explained before.

The L2 gradient flow equation is now written as the Cauchy-Riemann equation∂J = 0 associated to J . Its solutions are holomorphic curves, but here the situationdiffers drastically from the one dealt by Gromov in [19]: domains necessarily need tobe punctured Riemann surfaces. The behavior of these curves near their ends, undera certain finite-energy condition introduced by Hofer, allowed for Morse-homologicalconstructions, giving rise to Contact Homology and Symplectic Field Theory [9].

3. Global surfaces of section

As explained in the introduction, Poincaré constructed annulus-like global surfacesof section in the context of the PCR3BP. Another early and powerful existenceresult is the following.

Theorem 3.1 (Birkhoff [6]). The geodesic flow on the unit tangent bundle of anypositively curved Riemannian metric on S2 admits a global surface of section.

In fact, we can be more precise. Consider an embedded closed geodesic γ(t) oflength L in a positively curved Riemannian 2-sphere, parametrized with unit speed.Choose an ambient orientation and for each t ∈ R/LZ let n(t) be the unit vector atγ(t) such that {γ(t), n(t)} is a positive orthonormal frame. The Birkhoff annulusassociated to γ is

(16) Aγ = {cos θ γ(t) + sin θ n(t) | t ∈ R/LZ, θ ∈ [0, π]}.This is an embedded annulus in the unit sphere bundle, and Birkhoff showed that itis a global surface of section whenever the Gaussian curvature is positive everywhere.


Birkhoff’s proof is very specific to geodesic flows and does not shed much light intothe general existence problem.

A very general theory of global surfaces of section exists, for arbitrary flows on3-manifolds. It is the outcome of the work of many mathematicians during theXX century, including Schwartzman [34], Fried [16], Ghys [18] and Sullivan [35].However, most of the statements make use of dynamical hypotheses which are veryhard to be checked. This in contrast to Theorem 3.1.

Here we focus on existence statements based on the theory developed by Hofer,Wysocki and Zehnder (HWZ) since they resonate with Birkhoff’s statement in thesense that their hypotheses are quite concrete and geometric.

Consider R4 with coordinates (x1, y1, x2, y2) and its standard symplectic formω0 = dx1 ∧ dy1 + dx2 ∧ dy2. Let C ⊂ R4 be a compact convex set with strictlyconvex smooth boundary M = ∂C: this means that the boundary is smooth andhas positive sectional curvatures everywhere. Note that any Hamiltonian having Mas a regular level will define a smooth flow on M without stationary points, andanother choice of such Hamiltonian will change this flow only by reparametrizing it.Hence, one can talk about Hamiltonian dynamics on M up to time-parametrization.

The first and main result is the following remarkable statement.

Theorem 3.2 (HWZ [24]). Hamiltonian dynamics on strictly convex hypersurfacesinside (R4, ω0) always admit disk-like global surfaces of section.

Theorem 3.2 is proved using holomorphic curves. Consider M = ∂C the strictlyconvex smooth boundary of the convex body C, and the 1-form

λ0 =1

2

2∑j=1

xjdyj − yjdxj

which is a primitive of ω0. We denote α = ι∗λ0 where ι : M ↪→ R4 is the inclusion.We parametrize Hamiltonian dynamics on M by the unique vector field X satisfying

dα(X, ·) = 0 α(X) = 1

which is called the Reeb vector field.The kernel ξ = kerα is a contact structure. We represent the symplectization

of (M, ξ) as the symplectic manifold (R ×M,d(eaα)) where the R-coordinate isdenoted by a. We view α, X and ξ as R-invariant objects in the symplectization.One checks that dα turns ξ into a symplectic vector bundle.

Choose a complex structure J : ξ → ξ such that dα(·, J ·) > 0. Hofer considersthe almost complex structure J on R×M defined by

J : ∂a 7→ X, J |ξ = J.

The proof in [24] is based on the construction of finite-energy holomorphic planes in(R×M, J). These are smooth maps

u = (a, u) : (C, i)→ (R×M, J)

satisfying

(17) ∂(u) =1

2(du+ J(u) ◦ du ◦ i) = 0.


and a finite-energy condition introduced by Hofer [20]. HWZ construct such a planeasymptotic to a periodic trajectory x : R/TZ→M of X in the sense that

lims→+∞

u(e2π(s+it)) = x(Tt) in C∞(R/Z,M).

The trajectory x has very special properties from dynamical and contact topologicalpoints of view: its Conley-Zehnder index is equal to 3 and its self-linking number isequal to −1. Using this information, results from [22] tell us that the M -componentu : C→ M is an embedding transverse to X. The elliptic nature of (17) and theasymptotic behavior established in [21] allow for a Fredholm theory [23], whichis used by HWZ to view such a plane as one leaf of a local foliation in M \ x(R).Finally, one needs compactness results for punctured holomorphic curves to obtaina foliation on the whole of M \ x(R) by planes transverse to X. This transversalityand the positivity of the linearized dynamics at x are the reason why each leaf inthis foliation will be the interior of a disk-like global surface of section.

The reader might wonder where is the strict convexity of M used in HWZ’sargument. The answer is: both in the construction of the first holomorphic planeand in the passage from local to global foliations. The idea is that when one ofthese steps fail, the compactness results will produce other holomorphic curveswhich are asymptotic to periodic trajectories with Conley-Zehnder index less than 3.However, HWZ show that strict convexity forces the periodic Reeb trajectories tohave Conley-Zehnder index at least equal to 3, a property called dynamical convexityby HWZ.

At this point HWZ were able to obtain the following remarkable corollary.

Theorem 3.3 (HWZ [24]). Strictly convex energy levels in (R4, ω0) admit eithertwo or infinitely many periodic trajectories.

To prove this, note that the return map to the disk provided by Theorem 3.2preserves an area form with finite total area. Brouwer’s translation theorem gives afixed point, corresponding to a second periodic trajectory. Once this fixed point isremoved from the disk, we are left with a return map on an open annulus. Frank’sresults [13, 14] imply that there are either none or infinitely many periodic points.

In the search of finer global structures of the flow, one might ask if there aremore global surfaces of section on strictly convex energy levels. One might also askif non-convex levels admit global sections. The statements below were proved usingthe methods of HWZ.

Theorem 3.4 ([27, 28]). A periodic orbit on a strictly convex energy level bounds adisk-like global surface of section if, and only if, it is unknotted and has self-linkingnumber −1.

Theorem 3.5 ([29, 30]). A periodic orbit on a star-shaped energy level boundsa disk-like global surface of section if it is unknotted, its self-linking number is−1, its Conley-Zehnder index is at least 3, and it is linked to all periodic orbitswith transverse rotation number equal to 1. Conversely, these assumptions areC∞-generically necessary.

Excellent introductions to pseudo-holomorphic curve theory in symplectizationsand symplectic cobordisms can be found in the book [1] by Abbas and the book [36]by Wendl.


4. Quick introduction to the PCR3BP

We end this note by giving some details on this which is a central problem inCelestial Mechanics. This also serves as an excuse to point towards a different proofof a result from [7]; see Theorem 4.2. An excellent introduction to the PCR3BPare the books [15] by Frauenfelder and van Koert, and [17] by Geiges. The bookof Frauenfelder and van Koert also serves as an introduction to holomorphic curvemethods and their use in Celestial Mechanics.

At first one considers three massive bodies, their positions and masses beingdenoted by z1, z2, z3 and m1,m2,m3 respectively. According to Newton’s Law ofGravitation, motion is described by the following system of second order ODEs

z1 = m2z2 − z1|z2 − z1|3

+m3z3 − z1|z3 − z1|3

(18)

z2 = m1z1 − z2|z1 − z2|3

+m3z3 − z2|z3 − z2|3

(19)

z3 = m1z1 − z3|z1 − z3|3

+m2z2 − z3|z2 − z3|3

(20)

in suitably normalized units. Setting m3 = 0 in (18)-(19) one obtains the restrictedthree-body problem: the first two particles (primaries) move independently of thethird particle according to the two-body problem, and the third particle (masslesssatellite) moves according to (20). Requiring that all particles move on a plane oneobtains the planar problem; we make this assumption here, and from now on the zkbelong to the complex plane C. The relative position of the primaries ζ = z2 − z1solves Kepler’s equation ζ = −(m1 +m2)ζ/|ζ|3 and the adjective circular refers tothe case when ζ describes a circle. The PCR3BP is then the problem of describingthe movement of the satellite.

One can see (20) as a non-autonomous non-linear second order differential equationfor z3(t), which may lead the reader to think that it is intractable. However, a well-known miracle happens: in a certain non-inertial coordinate system the equationsof motion not only retain their Hamiltonian form, but the Hamiltonian functionturns out to be time-independent!

More precisely, consider a inertial coordinate system where the center of massrests at the origin. Since ζ describes circular motion, we find ω 6= 0 such that

(21) z1(t) = r1eiωt z2(t) = −r2eiωt

for some constants r1, r2 > 0. The condition on the center of mass reads m1r1 −m2r2 = 0, and from Kepler’s equation we extract the identity

(r1 + r2)3ω2 = m1 +m2.

There is no loss of generality to assume that the angular velocity ω is positive. Inthe rotating coordinate system with angular velocity ω, the position q(t) of thesatellite relative to the second primary is defined by

z3(t) = (q(t)− r2)eiωt

from where it follows, together with (20) and (21), that q(t) solves

(22) q + 2iωq − ω2(q − r2) = −m1q − r1 − r2|q − r1 − r2|3

−m2q

|q|3.


The first primary rests at the point r1 + r2 and the second primary rests at theorigin.

It is convenient to introduce the mass ratio

µ =m1

m1 +m2∈ [0, 1]

because if ω and m1 + m2 > 0 are fixed once and for all, then all constantsm1,m2, r1, r2 become functions of µ. Finally we set

p = q + iω(q − r2)

and consider the Hamiltonian

(23) Hµ(q, p) =1

2|p|2 + ω 〈q − r2, ip〉 −

m1

|q − r1 − r2|− m2

|q|where 〈·, ·〉 denotes the euclidean inner-product. The miracle that materializes infront of our eyes is that (22) is equivalent to Hamilton equations

q = ∇pHµ p = −∇qHµ

where the surprising fact is that this is an autonomous Hamiltonian system. As iswell known, the value of Hµ is preserved along trajectories:

d

dtHµ = 〈∇qHµ, q〉+ 〈∇pHµ, p〉 = 〈∇qHµ,∇pHµ〉 − 〈∇pHµ,∇qHµ〉 = 0.

This is one reason why Hµ may be called energy, and we can say that energy ispreserved. The value of c defined by the identity

(24) − 1

2c = Hµ

is historically called the Jacobi constant.Let us understand a bit the geometry behind some of the energy levels of Hµ.

The function Hµ is unbounded from above and below. Completing the squaresin (23) we get

Hµ(q, p) =1

2|q|2 − Uµ(q)

where

(25) Uµ(q) =1

2|ω(q − r2)|2 +

m1

|q − r1 − r2|+m2

|q|is the effective potential. The projection to configuration space, i.e. to the q-plane,of the sublevel set {Hµ ≤ −c/2} coincides with the superlevel set {Uµ ≥ c/2}.

If c ∼ +∞ then {Uµ ≥ c/2} has three connected components: two disk-likeregions around the positions 0 and r1 + r2 of the primaries, and an unboundedcomponent given as the complement of a large disk-like region. These are the Hill’sregions. Their boundaries are called ovals of zero velocity for the following reason:if the energy of a satellite inside a bounded Hill region in {Uµ ≥ c/2} is constrainedby Hµ ≤ −c/2 then the satellite can not leave this region, and can only touch itsboundary with zero velocity q = 0 and only if its energy is precisely −c/2. In otherwords, the satellite does not have enough energy to “escape” the influence of thecorresponding primary.

This is the situation when the Jacobi constant −c/2 increases from −∞ untilc/2 reaches the highest critical critical value c1/2 of Uµ. The only critical point ofUµ at this level lies strictly between the primaries, and we see the two boundedHill regions getting “glued” together. For energy −c/2 = −c1/2 + ε slightly higher


than −c1/2 the satellite may get arbitrarily close to both primaries, i.e. now thereis enough energy to escape. If the Jacobi constant decreases to −∞ then c/2 willcross three more critical values of Uµ corresponding to further modification of thetopology of the Hill regions.

Throughout the remaining of this section we stick to values c > c1 and focus on theHill region around the origin. We make normalization assumptions ω = m1+m2 = 1,without loss of generality. It follows that m1 = r2 = µ, m2 = r1 = 1− µ and Hµ iswritten as

Hµ =|p|2

2+ 〈ip, q − µ〉 − µ

|q − 1|− 1− µ|q|

the unique parameter being µ. Simple calculations show that c1 → 3 both whenµ→ 0+ and when µ→ 1−.

The method of Levi-Civita [31], which we shall now describe, regularizes collisionswith the primary at the origin. We introduce new complex coordinates u, v by

q = 2v2 p = −uv.

This amounts to a two-to-one transformation of 4-space which is symplectic up to aconstant factor. In fact,

dq1 ∧ dp1 + dq2 ∧ dp2 = Re [dq ∧ dp]

= −2 Re[d(v2) ∧ d

( uv

)]= −2 Re

[2vdv ∧

(vdu− udv

v2

)]= −4 Re [dv ∧ du]

= 4 (du1 ∧ dv1 + du2 ∧ dv2) .

It follows that Hamiltonian flows on (q, p)-space correspond to Hamiltonian flowson (u, v)-space up to a constant time-reparametrization.

We then reparametrize the Hamiltonian flow of Hµ on the level {Hµ = − c2} as

the Hamiltonian flow of Kµ,c = 12 |q|(Hµ+ c

2 ) = |v|2(Hµ+ c2 ) on the level {Kµ,c = 0}.

Writing Kµ,c in terms of u, v we get

(26) Kµ,c(u, v) =1

2|u|2 + 2|v|2 〈u, iv〉 − µ Im[uv]− 1− µ

2− µ |v|2

|2v2 − 1|+c

2|v|2

The upshot here is that |q−1| = |2v2−1| > 0 as long as c > c1, hence the componentΣµ,c of {Kµ,c = 0} corresponding to our chosen Hill region is a smooth hypersurfaceof R4 where the Hamiltonian flow is smooth. At this point it is worth pointing outthe following nice statement.

Theorem 4.1 (Albers, Fish, Frauenfelder, Hofer and van Koert [2]). For everyc > 3 there exists µ0(c) ∈ (0, 1) such that if µ > µ0(c) then Σµ,c is strictly convex, inthe sense that its sectional derivatives are strictly positive. In particular Theorem 3.2applies to give disk-like global surfaces of section.

Consider a new set of coordinates

u = u v =√c v


which are again symplectic up to a constant factor, and write

(27) Kµ,c(u, v) =1

2|u|2 + 2

|v|2 〈u, iv〉c√c

− µ Im[uv]√c− 1− µ

2− µ |v|2

c| 2c v2 − 1|+

1

2|v|2

in terms of these variables. If c→ +∞ then Kµ,c(u, v) converges in C∞ to1

2|u|2 +

1

2|v|2 − 1− µ

2

which is the Hamiltonian of two uncoupled harmonic oscillators with equal frequen-cies. In particular, the flow converges to a periodic flow whose trajectories are theHopf fibers in the sphere of radius

√1− µ. In fact, the C∞-converge is uniform

when µ ranges on a compact subset of [0, 1). This simple observation and standardarguments can be used to prove

Theorem 4.2 (Conley [7]). If µ ∈ (0, 1) is fixed and the Jacobi constant is largeenough then there are annulus-like global surfaces of section for the Hamiltonianflow on the component of the energy level {Kµ,c = 0} corresponding to the Hillregion around the origin. The return map to these annuli satisfy the twist conditionof the Poincaré-Birkhoff Theorem, in particular there are infinitely many periodicorbits for the PCR3BP in these situations.

References

[1] C. Abbas, An introduction to compactness results in symplectic field theory. Springer, Heidel-berg, 2014. viii+252 pp. ISBN: 978-3-642-31542-8; 978-3-642-31543-5.

[2] P. Albers, J. W. Fish, U. Frauenfelder, H. Hofer, and O. van Koert. Global surfaces of sectionin the planar restricted 3-body problem. Archive for Rational Mechanics and Analysis, 204(2012), no.1, 273–284.

[3] V.I. Arnold, Mathematical methods of classical mechanics. Springer-Verlag, Berlin and NewYork, 1978.

[4] M. Audin and M. Damian, Morse theory and Floer homology. Translated from the 2010French original by Reinie Erné. Universitext. Springer, London; EDP Sciences, Les Ulis, 2014.xiv+596 pp. ISBN: 978-1-4471-5495-2; 978-1-4471-5496-9; 978-2-7598-0704-8.

[5] G. D. Birkhoff, Proof of Poincaré’s last geometric theorem, Trans. Amer. Math. Soc., 14(1913), 14–22.

[6] G. D. Birkhoff. Dynamical Systems, (Colloq. Publ., Am. Math. Soc., vol. 9) Providence, RI.:Amer. Math. Soc. 1927.

[7] C. Conley. On some new long periodic solutions of the plane restricted three body problem.Commun. Pure Appl. Math. 16, (1963) pp. 449–467.

[8] C. Conley and E. Zehnder, The Birkhoff-Lewis fixed point theorem and a conjecture of V.I.Arnold. Invent. Math., 73:33–49, 1983.

[9] Y. Eliashberg, A. Givental, H. Hofer. Introduction to Symplectic Field Theory. Geom. Funct.Anal., Special volume, Part II (2000), 560-673.

[10] A. Floer, Symplectic fixed points and holomorphic spheres. Comm. Math. Phys. 120 (1989),no. 4, 575–611.

[11] A. Floer, Morse theory for Lagrangian intersections. J. Differential Geom. 28 (1988), no. 3,513–547.

[12] A. Floer, The unregularized gradient flow of the symplectic action. Comm. Pure Appl. Math.41 (1988), no. 6, 775–813.

[13] J. Franks. Recurrence and fixed points of surface homeomorphisms. Erg. Th. Dyn. Sys., 8,(1988), 99–107.

[14] J. Franks. Geodesics on S2 and periodic points of annulus homeomorphisms, Invent. Math.108 (1992), no. 2, 403–418.

[15] U. Frauenfelder and O. van Koert, The Restricted Three-Body Problem and HolomorphicCurves (Pathways in Mathematics). Birkhäuser Basel (2018).

[16] D. Fried, The geometry of cross sections to flows, Topology, 21 (1982), 353–371.


[17] H. Geiges, The Geometry of Celestial Mechanics (London Mathematical Society StudentTexts). Cambridge: Cambridge University Press (2016).

[18] E. Ghys, Right-handed vector fields & the Lorenz attractor, Japan. J. Math. 4, 47–61 (2009).[19] M. Gromov. Pseudo-holomorphic curves in symplectic manifolds. Invent. Math. 82 (1985),

307–347.[20] H. Hofer. Pseudoholomorphic curves in symplectisations with application to the Weinstein

conjecture in dimension three. Invent. Math. 114 (1993), 515-563.[21] H. Hofer, K. Wysocki and E. Zehnder. Properties of pseudoholomorphic curves in symplecti-

sations I: Asymptotics. Ann. Inst. H. Poincaré Anal. Non Linéaire 13 (1996), 337–379.[22] H. Hofer, K. Wysocki and E. Zehnder. Properties of pseudoholomorphic curves in symplecti-

sations II: Embedding controls and algebraic invariants. Geom. Funct. Anal. 5 (1995), no. 2,270–328.

[23] H. Hofer, K. Wysocki and E. Zehnder. Properties of pseudoholomorphic curves in sym-plectizations III: Fredholm theory. Topics in nonlinear analysis, Birkhäuser, Basel, (1999),381–475.

[24] H. Hofer, K. Wysocki and E. Zehnder. The dynamics on three-dimensional strictly convexenergy surfaces. Ann. of Math. 148 (1998), 197–289.

[25] H. Hofer, K. Wysocki and E. Zehnder. Finite energy foliations of tight three-spheres andHamiltonian dynamics. Ann. Math 157 (2003), 125–255.

[26] H. Hofer and E. Zehnder, Symplectic Invariants and Hamiltonian Dynamics. Birkhäuser,Basel, 1994.

[27] U. Hryniewicz. Fast finite-energy planes in symplectizations and applications. Trans. Amer.Math. Soc. 364 (2012), 1859–1931.

[28] U. Hryniewicz. Systems of global surfaces of section for dynamically convex Reeb flows on the3-sphere, Journal of Symplectic Geometry 12, 4, (2014).

[29] U. Hryniewicz, J. E. Licata and Pedro A. S. Salomão. A dynamical characterization ofuniversally tight lens spaces. arXiv:1306.6617v2.

[30] U. Hryniewicz and Pedro A. S. Salomão. On the existence of disk-like global sections for Reebflows on the tight 3-sphere. Duke Math. J. 160 (2011), no. 3, 415–465.

[31] T. Levi-Civita, Sur la régularisation du problème des trois corps. Acta Math. 42(1), 99–144(1920).

[32] H. Poincaré. Sur un théorème de géométrie. Rend. Circ. Mat. Palermo, 33 (1912), 375–407.[33] D. Salamon, Lectures on Floer homology. Symplectic geometry and topology (Park City, UT,

1997), 143?229, IAS/Park City Math. Ser., 7, Amer. Math. Soc., Providence, RI, 1999.[34] S. Schwartzman, Asymptotic cycles, Ann. of Math. (2) 66 (1957), 270–284.[35] D. Sullivan, Cycles for the dynamical study of foliated manifolds and complex manifolds,

Invent. Math., 36 (1976), 225–255.[36] C. Wendl, Holomorphic Curves in Low Dimensions. Springer International Publishing, 2018.

xiii+294 pp. ISBN: 978-3-319-91369-8

Universidade Federal do Rio de Janeiro, Departamento de Matematica Aplicada,Av. Athos da Silveira Ramos 149, Rio de Janeiro RJ, Brazil 21941-909.Email address: [email protected]

holomorphic curves and celestial mechanics

Documents